Q.A galvanometer of resistance $10\ \Omega$ that gives maximum (full-scale) deflection for a current of $1\ \text{mA}$ is to be converted into a multirange current meter (ammeter) reading $10\ \text{mA}$, $100\ \text{mA}$ and $1\ \text{A}$. Three shunt resistors $S_1$, $S_2$ and $S_3$ are joined in series with one another and this chain is connected across the galvanometer (an Ayrton-shunt arrangement). For the $10\ \text{mA}$ range the whole chain $S_1+S_2+S_3$ acts as the shunt and the galvanometer alone forms the other branch; for the $100\ \text{mA}$ range $S_2+S_3$ is the shunt while $S_1$ is in series with the galvanometer; for the $1\ \text{A}$ range only $S_3$ is the shunt while $S_1+S_2$ is in series with the galvanometer. Find $S_1$, $S_2$ and $S_3$.

Concept & formula

In a shunted ammeter the galvanometer (resistance $G$, full-scale current $I_g$) is in parallel with a shunt of resistance $S_{\text{sh}}$. The remaining current $(I-I_g)$ flows through the shunt, and the two parallel branches share the same potential difference:

Here $G=10\ \Omega$, $I_g=10^{-3}\ \text{A}$, and the shunt/series split changes with the selected terminal.

Step 1 — $10\ \text{mA}$ range (shunt $=S_1+S_2+S_3$, galvanometer alone)

Step 2 — $100\ \text{mA}$ range (shunt $=S_2+S_3$, series $=G+S_1$)

$10^{-3}(10+S_1)=0.099\left(\tfrac{10}{9}-S_1\right)\ \Rightarrow\ 0.01+10^{-3}S_1=0.11-0.099S_1\ \Rightarrow\ 0.1\,S_1=0.1,$ …

Method: Solving a Multi-Range (Ayrton) Shunt Ammeter

This method applies to converting a galvanometer into a multi-range ammeter using a chain of shunt resistors tapped at different points (an Ayrton shunt), where each range uses a different split between the "shunt" portion (parallel to the galvanometer) and the "series" portion (in series with it).

Steps

Step 1: For each range, identify the shunt portion and the series portion

Read the tap/switch description for that range carefully: which resistors of the chain are now in parallel with the galvanometer (the shunt), and which are in series with the galvanometer coil. This split is DIFFERENT for every range and is given by how the Ayrton chain is tapped.

Step 2: Write the equal-voltage condition for that range

The galvanometer branch and the shunt branch are in parallel, so they share the same potential difference. At full-scale deflection, the galvanometer carries its rated current $I_g$, and the rest of the range current flows through the shunt:

Step 3: Repeat for every range to build a system of equations

Each range gives one equation in terms of the unknown shunt resistors. Because the shunt/series split changes between ranges, later equations will contain a mix of already-solved and still-unknown resistors.

Step 4: Solve sequentially, easiest range first …